14 (b) Considr a diffrntial lmnt which is mad up of a thin cylindr of innr radius r and outr radius r + dr and lngth L. Its contribution to th rsistanc of th systm is givn by ρ dl ρ dr dr = = A π rl whr A= π rl is th ara normal to th dirction of currnt flow. Th total rsistanc of th systm bcoms b ρdr ρ b R = = ln a πrl πl a 6.6 Concptual Qustions 1. Two wirs A and B of circular cross-sction ar mad of th sam mtal and hav qual lngths, but th rsistanc of wir A is four tims gratr than that of wir B. Find th ratio of thir cross-sctional aras.. From th point of viw of atomic thory, xplain why th rsistanc of a matrial incrass as its tmpratur incrass. 3. Two conductors A and B of th sam lngth and radius ar connctd across th sam potntial diffrnc. Th rsistanc of conductor A is twic that of B. To which conductor is mor powr dlivrd? 6.7 Additional Problms Currnt and Currnt Dnsity 9 A sphr of radius 10 mm that carris a charg of 8 nc = 8 10 C is whirld in a circl at th nd of an insulatd string. Th rotation frquncy is 100π rad/s. (a) What is th basic dfinition of currnt in trms of charg? (b) What avrag currnt dos this rotating charg rprsnt? (c) What is th avrag currnt dnsity ovr th ara travrsd by th sphr? 6.7. Powr Loss and Ohm s Law A 1500 W radiant hatr is constructd to oprat at 115 V. 6-14

15 (a) What will b th currnt in th hatr? [Ans. ~10 A] (b) What is th rsistanc of th hating coil? [Ans. ~10 Ω] (c) How many kilocaloris ar gnratd in on hour by th hatr? (1 Calori = 4.18 J) Rsistanc of a Con A coppr rsistor of rsistivity ρ is in th shap of a cylindr of radius b and lngth L 1 appndd to a truncatd right circular con of lngth L and nd radii b and a as shown in Figur Figur (a) What is th rsistanc of th cylindrical portion of th rsistor? (b) What is th rsistanc of th ntir rsistor? (Hint: For th taprd portion, it is ncssary to writ down th incrmntal rsistanc dr of a small slic, dx, of th rsistor at an arbitrary position, x, and thn to sum th slics by intgration. If th tapr is small, on may assum that th currnt dnsity is uniform across any cross sction.) (c) Show that your answr rducs to th xpctd xprssion if a = b. (d) If L 1 = 100 mm, L = 50 mm, a = 0.5 mm, b = 1.0 mm, what is th rsistanc? Currnt Dnsity and Drift Spd (a) A group of chargs, ach with charg q, movs with vlocity v. Th numbr of particls pr unit volum is n. What is th currnt dnsity J of ths chargs, in magnitud and dirction? Mak sur that your answr has units of A/m. (b) W want to calculat how long it taks an lctron to gt from a car battry to th startr motor aftr th ignition switch is turnd. Assum that th currnt flowing is115 A, and that th lctrons travl through coppr wir with cross-sctional ara 31. mm and lngth 85.5 c m. What is th currnt dnsity in th wir? Th numbr dnsity of th 8 3 conduction lctrons in coppr is /m. Givn this numbr dnsity and th currnt dnsity, what is th drift spd of th lctrons? How long dos it tak for an 6-15

16 6 lctron starting at th battry to rach th startr motor? [Ans: A/m, m/s, 5.5 min.] Currnt Sht A currnt sht, as th nam implis, is a plan containing currnts flowing in on dirction in that plan. On way to construct a sht of currnt is by running many paralll wirs in a plan, say th yz -plan, as shown in Figur 6.7.(a). Each of ths wirs carris currnt I out of th pag, in th j ˆ dirction, with n wirs pr unit lngth in th z-dirction, as shown in Figur 6.7.(b). Thn th currnt pr unit lngth in th z dirction is ni. W will us th symbol K to signify currnt pr unit lngth, so that K = nl hr. Figur 6.7. A currnt sht. Anothr way to construct a currnt sht is to tak a non-conducting sht of charg with fixd charg pr unit ara σ and mov it with som spd in th dirction you want currnt to flow. For xampl, in th sktch to th lft, w hav a sht of charg moving out of th pag with spd v. Th dirction of currnt flow is out of th pag. (a) Show that th magnitud of th currnt pr unit lngth in th z dirction, K, is givn by σ v. Chck that this quantity has th propr dimnsions of currnt pr lngth. This is in fact a vctor rlation, K (t) = σ v( t), sinc th sns of th currnt flow is in th sam dirction as th vlocity of th positiv chargs. (b) A blt transfrring charg to th high-potntial innr shll of a Van d Graaff acclrator at th rat of.83 mc/s. If th width of th blt carrying th charg is 50 cm and th blt travls at a spd of 30 m /s, what is th surfac charg dnsity on th blt? [Ans: 189 µc/m ] Rsistanc and Rsistivity A wir with a rsistanc of 6.0 Ω is drawn out through a di so that its nw lngth is thr tims its original lngth. Find th rsistanc of th longr wir, assuming that th 6-16

ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating

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